# Hessian Matrix convex, concave, or neither?

I want to find out in what range a certain function is convex. In order to find out, I calculated partial derivatives and set up a Hessian matrix. As the Hessian still has variables in it, I am not sure about the interpretation. I would like to work with the method of principal minors.

I want to make use of this theorem:

Theorem: $$f(x,y)$$ is convex if and only if its $$n \times n$$ Hessian matrix is positive semidefinite for all possible values of $$(x,y)$$. The Hessian is positive definite if and only if its $$n=2$$ leading principal minors are positive.

My $$2\times 2$$ matrix: $$\begin{bmatrix} 6x+4 & 7855\\7855 & 2\end{bmatrix}$$

1. $$6x+4$$
2. The determinant of the full matrix, $$(6x+4)2 - 7855^2$$.
For what values of $$x$$ are both of these positive? Those are the values of $$x$$ for which your function is (strictly) convex. (Note that in principle you only need to check the determinant, because the bottom-right entry is already positive ($$2$$) and you can permute the rows and columns so that it becomes the top-right leading principal minor instead of $$6x+4$$).